I have also come to chennai like you just 10 days before from Delhi and facing the same problem. But I have tasted the water of Chennai I didn't find it salty but yes it's very hard and rash for hair.
Do not believe much in shampoos and other things, afterall you are washing your hair with the same hard water. One thing I would like to mention you can do one thing after your bath take 2-3 mugs of drinking water and rinse your hair with it, it will help to remove the excessive hardness.
Second thing is take care of your diet, because nature has made us to fight in any circumstance against bad things. Take lot of juices and eat healthy and take vitamin capsules 4-5 every week to compensate the necessary requirements of a body.
Last but not least do not go directly in the sun always use umberella while going out, the pinching heat of Chennai is also a major cause of hair fall and other skin problems like tanning.
And use coconut oil , apply in the night twice a week , it will help a lot. I am also very pissed off with the harsh climate of Chennai but no option bro so just fight against it.